Regular Maps of High Density

Abstract

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a simple graph with ratio between vertex-degree and number of vertices strictly exceeding 1 2 . We conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves x n +y n +z n = 0, excepting the one corresponding to the tetrahedron.